Acoustic building infiltration measurement system

ABSTRACT

Systems and methods of detecting and identifying a leak from a container or building. Acoustic pressure and velocity are measured. Acoustic properties are acquired from the measured values. The acoustic properties are converted to infiltration/leakage information. Nearfield Acoustic Holography (NAH) may be one method to detect the leakages from a container by locating the noise sources.

STATEMENT OF GOVERNMENT INTEREST

The United States Government has rights in the invention describedherein pursuant to Contract No. DE-AC02-06CH11357 between the UnitedStates Department of Energy and UChicago Argonne, LLC, as operator ofArgonne National Laboratory.

FIELD OF THE INVENTION

The present invention generally relates to leakage or infiltrationdetection. More specifically, the present invention relates to systemsand methods of identifying and converting acoustic properties toinfiltration or leakage information.

BACKGROUND OF THE INVENTION

Building air infiltration, the uncontrolled leakage of air in and out ofbuildings, accounts for over 30% of total heating load of commercialbuildings in the US. Measuring the infiltration in commercial buildingsis difficult because of the physical size and measurements cannot bemade during construction which means sealing the building becomes moredifficult and expensive.

Two methods for measuring building infiltration are currently used:pressurization testing and tracer gas testing. Pressurization testingworks by pressurizing an entire building or building section with fansand measuring the air flow through and pressure difference across theenvelope. This method works well for small buildings but is impracticalfor large buildings. Further, the entire building envelope must beconstructed before the method can be used. The tracer gas method worksby introducing a gas into a room at a known rate and measuring theamount of time for the gas concentration to be reduced throughinfiltration and exfiltration. The tracer technique can take a very longtime (especially on buildings with low infiltration), is more difficultto apply in large buildings, and also needs a completed buildingenclosure to utilize.

Others have attempted to correlate sound transmission loss toinfiltration rates with limited success. While correlations do exist,they are specific to the exact details of wall construction andinfiltration location so one cannot develop correlations on one buildingand use them on another and hence the method cannot be used as a generalmethod for measuring infiltration. Further, there is no general methodon how to estimate the correlations without actual measurement of thebuilding (i.e. from drawings and computer models).

Leakages in a gas-storing containers and air-conditioned buildings areserious safety and economic problem to the society. However, the presentleakage detection methods such as smoke stick method are not veryeffective and involves a lot of human resource.

SUMMARY OF THE INVENTION

One embodiment of the invention relates to A method of detectingleakage. Sound is emitted from an electroacoustic transducer positionedwithin a structure. Internal pressure of the emitted sound within thestructure is measured. Sound external to the structure is detected.External pressure and external velocity of the detected sound externalto the device is measured. A leak in the structure is determined.

Another implementation relates to a system for detecting leakage. Anelectroacoustic transducer is configured to emit an acoustic wave havinga phase, frequency and amplitude, the transducer in communication with acomputer to provide the phase, frequency, and amplitude of the emittedacoustic wave. An acoustic transducer system is configured to detectacoustic pressure and acoustic velocity

Another implementation relates to a computer system for leakagedetection. An electroacoustic transducer is configured to emit anacoustic wave having a phase, frequency and amplitude, the transducer incommunication with a computer to provide the phase, frequency, andamplitude of the emitted acoustic wave. A microphone array is configuredto receive the acoustic wave, the microphone array in communication withthe computer providing acoustic velocity and acoustic pressure data tothe computer. The computer includes a processor and a tangiblecomputer-readable medium operatively connected to the processor andincluding computer code configured to determine from the array acousticvelocity and pressure data a pressure associated with each of aplurality of points.

Additional features, advantages, and embodiments of the presentdisclosure may be set forth from consideration of the following detaileddescription, drawings, and claims. Moreover, it is to be understood thatboth the foregoing summary of the present disclosure and the followingdetailed description are exemplary and intended to provide furtherexplanation without further limiting the scope of the present disclosureclaimed.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and other objects, aspects, features, and advantages ofthe disclosure will become more apparent and better understood byreferring to the following description taken in conjunction with theaccompanying drawings, in which:

FIG. 1A Illustrates the geometry of a crack, such as in buildinginfiltration, that is H in height, L in length, and D in depth; FIG. 1Billustrates geometry for studying flow into the inlet of the crack ofheight H of FIG. 1A.

FIG. 2 is a basic diagram of the ABIMS system. Transmission of soundwaves is measured with a nearfield acoustic holography array. These dataare converted to infiltration estimates.

FIGS. 3A-B illustrates a traversing single microphone setup for (a)qualification test (b) building block measurement; FIG. 3C-D illustratethe two inserts used in the described experiments to simulate a leakagearea of a building.

FIGS. 4A-C illustrates the results of a qualification test for variousNAH algorithms, contours, algorithms, contours of p_(s) for the inputfrequency f=5004 Hz for (A) FFT, (B) BEM and (C) ESM.

FIGS. 5A-C are graphs of reconstructed absolute acoustic pressure usingvarious algorithms, insert 1, f=5004 Hz for (A) FFT, (B) BEM and (C)ESM.

FIGS. 6A-C are graphs of reconstructed absolute acoustic pressure usingvarious algorithms insert 11, f=5004 Hz3 for (A) FFT, (B) BEM and (C)ESM.

FIGS. 7A-C are graphs of reconstructed absolute particle velocity usingvarious algorithms insert I, f=5004 Hz for (A) FFT, (B) BEM and (C) ESM.

FIGS. 8A-C are graphs of reconstructed absolute particle velocity usingalgorithms, insert II, f=5004 Hz for (A) FFT, (B) BEM and (C) ESM

FIGS. 9A-C are graphs of reconstructed absolute acoustic pressure usingvarious algorithms, insert I, f=317 Hz for (A) FFT, (B) BEM and (C) ESM.

FIGS. 10A-C are graphs of reconstructed absolute acoustic pressure usingvarious algorithms, insert II, f=317 Hz for (A) FFT, (B) BEM and (C)ESM.

FIGS. 11A-C are graphs of reconstructed absolute particle velocity usingvarious algorithms, insert I, f=317 Hz for (A) FFT, (B) BEM and (C) ESM.

FIG. 12A-C are graphs of reconstructed absolute particle velocity usingvarious algorithms, insert II, f=317 Hz for (A) FFT, (B) BEM and (C)ESM.

FIGS. 13A-B is a graph of acoustic pressure (13A) and particle velocity(13B) using ESM, insert II, f=1098 Hz.

FIGS. 14A-B are graphs of acoustic pressure (14A) and particle velocity(14B) using ESM-GCV, f=317 Hz; Dx=0:25 in

FIGS. 15A-C are graphs of reconstructed absolute particle velocitycalculated with reduced resolution (Dx=0:5 in) using various algorithms,insert I, f=317 Hz for (A) FFT, (B) BEM and (C) ESM.

FIG. 16 is an illustration of a computer implementation of anembodiment.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

In the following detailed description, reference is made to theaccompanying drawings, which form a part hereof. In the drawings,similar symbols typically identify similar components, unless contextdictates otherwise. The illustrative embodiments described in thedetailed description, drawings, and claims are not meant to be limiting.Other embodiments may be utilized, and other changes may be made,without departing from the spirit or scope of the subject matterpresented here. It will be readily understood that the aspects of thepresent disclosure, as generally described herein, and illustrated inthe figures, can be arranged, substituted, combined, and designed in awide variety of different configurations, all of which are explicitlycontemplated and made part of this disclosure.

One implementation provides methods and systems for implementing anAcoustic Building Infiltration Measurement System (ABIMS). ABIMS is adevice which uses acoustic methods for measuring building airinfiltration (or leakage). It should be appreciated that whether theconcern or reality of a structure is leakage from the structure to theoutside environment or infiltration into the structure from the outsideenvironment, the concepts described herein are applicable, although inthe context of this disclosure only leakage or infiltration may bedescribed for a given implementation. An ABIMS converts acousticproperties to leakage information. In one implementation, the presentinvention relates to a system for and methods of 1) measuring acousticpressure and velocity, 2) acquiring acoustic properties such as acousticimpedance, or porous media properties (tortuosity, complex density,complex compressibility, flow resistivity, viscous permeability, etc)from the pressure and velocity measurements, and 3) converting theacoustic properties to infiltration/leakage information.

In building science, infiltration is typically referred to as a functionof an infiltration flow coefficient and a pressure difference modifiedby a power. Specifically, it is typical to express the relationshipbetween infiltration airflow (denoted as Q with units of m³/s), and thepressure difference between the building interior and exterior (denotedas ΔP with units of Pa) by a power law of the form (Sherman, 1992):Q=C _(L)(ΔP)^(n).  (1)where C_(L) is the infiltration flow coefficient and n is theinfiltration exponent. Typically, both C and n depend upon the geometryof the crack or cracks through which the infiltration is occurring.Analytic expressions for C and n for Eqn. (1) are desired for somesimple geometries. Further determine the infiltration, it is necessaryto determine infiltration flow coefficient.

Measuring Acoustic Pressure and Velocity

It should be appreciated that several techniques can be utilized withthe above method to provide the measurement of acoustic pressure andvelocity. When velocity is referred to herein, such is the particlevelocity not the speed of sound. Systems for use in measuring theacoustic pressure include, but are not limited to: transducers,including pressure sensors such as electret, condenser, moving coil, orpiezoelectric microphones, and piezoelectric microphones. Systems foruse in measuring the acoustic particle velocity include, but are:conversion of sound pressure measured by pressure sensors as is used intwo microphone sound intensity probes, ribbon microphones, MEMS basedprobes such as that sold by Microflow Technologies®, particle imagevelocimetry, and Doppler measurement systems such as Laser DopplerVibrometry (LDV) or Laser Doppler Anemometry (LDA).

One implementation of the present invention utilizes Nearfield AcousticHolography (“NAH”) to measure the acoustic pressure and velocity. Oncethe acoustic pressure and velocity are determined, that sound leakageinformation is converted to a static pressure flow equivalent usingbasic equations derived from fluid mechanics. From that, standardinfiltration measures are computed. In other implementations, the systemand methods can utilize any other technique that can measure both thepressure amplitude and particle velocity of sound at specific locationson the building enclosure surface.

Conformal NAH is based on Fourier transform and convolution, such asutilizing fast Fourier transform (“FFT”). These FFT based methods poseconstraints on the geometry and distribution of sources and transducers(microphones in most of the cases). These limitations created therequirement for modern NAH algorithms based on numerical methods tosolve the linear acoustic equations. NAH works based on the datameasured on a hologram surface to reconstruct the acoustic quantities inthe three dimensional domain using the boundary integral form ofHelmholtz equation. Therefore, a natural choice of numerical method wasthe boundary element method (BEM). Another simple yet powerful NAHalgorithm called equivalent source model (“ESM”) (also called wavesuperposition method) was used in many of the recent studies. Later, Baiintroduced a beamformer-like ESM algorithm in time domain and tested itexperimentally by prior researchers. Miller et al and Valdivia et alcompared the two algorithms (BEM and ESM) extensively. The BEM algorithmis usually considered as more accurate method. However, the ESMalgorithm is easier to implement and requires less CPU time compared toBEM. A hybrid method by combining the BEM algorithm with anotheralgorithm called Helmholtz least-squares (HELS) method was created by Wuto solve the Helmholtz integral equation. Possibilities of extending themeasurement surface using the analytic continuation was suggested byValdivia which reduces the measurement aperture restrictions.

The matrices resulting from the inverse boundary element method and theequivalent source model become very ill-conditioned as the system growslarger and larger. The ill-conditioned matrices usually amplify theerrors in the measurement when inverted because of the numericalsingularities. This error amplification is usually reduced usingregularization techniques such as Tikhonov regularization. Hansenprovided the details of all the modern methods of regularization.Williams provided a set of regularization methods that are particularlysuitable for NAH problems. While the direct regularization methods areused in most of the NAH studies, some of the researchers have adaptedthe iterative regularization methods in their studies mainly because theiterative methods can handle larger matrices with lesser memoryrequirement.

In one specific implementation, nearfield acoustic holography is usedwith a microphone. Conformal and computational methods of NAH have beenproven as successful tools to locate and quantify the vibro-acousticfields. As an inherently nonintrusive technique, the NAH method couldpossibly emerge as a potential alternative to the present leakagedetection methods. Three specific algorithms of NAH are discussedherein, but other algorithms may be used with various implementations.The primary NAH algorithms are Fourier transform (FFT) based NAH,Boundary element method (BEM) based NAH, and Equivalent source model(ESM) based NAH.

The following symbols and variables are used as set forth below:

-   γ Regularization parameter-   σ_(i) Singular values-   ω Angular velocity 2π f-   ψ_(S) Any parameter ψ on the source surface-   ψ_(H) Any parameter ψ on the hologram surface-   ψ_(V) Any parameter ψ on the virtual source surface-   ρ Density of medium-   ξ Local coordinates in the element A^(†)-   Pseudo-inverse of any matrix A c-   Speed of sound-   f Frequency-   f_(i) Filter factors-   G Free-space Green's function-   k Wave number-   k_(x), k_(y) Wave numbers in the x and y directions-   N_(i) Shape function-   P Acoustic pressure in frequency domain-   {tilde over (p)} Acoustic pressure in time domain-   {circumflex over (p)} Acoustic pressure in k, k_(x) and k_(y) domain-   q Source strength r Position vector-   t Time-   v_(x) Particle velocity component in a direction x x Global    coordinate

Preferred embodiments of the invention place an array of microphones onone side of the structure to be measured and an electroacoustictransducer, such as a loudspeaker on the other side. The speaker emitssingle frequency tones, multiple frequency tones, or sweeps offrequency/amplitude audio while simultaneously wirelessly communicatingphase/amplitude/frequency information. If single frequency tones areused, multiple measurements with tones at different frequencies must bemade. There is no fixed range of frequencies that must spanned by thetones or the sweep, but a range of 100 Hz to 5 kHz is generallypreferred to ensure there is sound close at fairly low audio frequenciesand fairly high audio frequencies This information is combined with theinformation picked up from the microphone array to measure the leakageof sound waves through the structure via a NearField Acoustic Holographysignal processing technique. The sound leakage information is convertedto a static pressure flow equivalent, using basic fluid mechanics, thenbe easily computed to provide infiltration measurements. Amplifiers andloudspeakers should be capable of producing frequencies below atamplitudes in excess of 60 dB at frequencies below 100 Hz and above 5kHz. Microphones should have a flat amplitude and phase response overthe 100 Hz to 5 kHz range. Microphones should be phase and amplitudematched as much as possible or have amplitude and phase calibration dataavailable to calibrate the microphone output before processing with theNAH algorithms.

If an acoustic pressure field {tilde over (p)}(r, t) obeys the waveequation

$\begin{matrix}{{{\Delta^{2}\overset{\sim}{p}} - {\frac{1}{c^{2}}\frac{\partial^{2}\overset{\sim}{p}}{\partial t^{2}}}} = 0} & (2)\end{matrix}$

in the space-time domain given by the position vector r and time t, thenthe nearfield acoustic holography (NAH) methods can be used to estimatethe propagation characteristics of the sound field. It is usually verydifficult to measure acoustic pressure on the source (S) surface.Therefore, the acoustic pressure {tilde over (p)}(r_(H), t) is usuallymeasured at on a hologram (H) plane which is very close to the sourcesurface. This measurement data is present at N discrete points on thehologram. The Fourier transform of {tilde over (p)}(r, t) is given by:p(r,ω)=∫_(31 ∞) ^(∞) {tilde over (p)}(r,t)e ^(inst) dt  (3)

The complex acoustic pressure field p is now in frequency domain.Applying Fourier transform on the wave equation [equation (2)] providesthe Helmholtz equation (also called reduced wave equation),Δ² p(r,ω)+k ² p(r,ω)=0  (4)

Here, k is wave number k=ω/c. From Euler's equation, particle velocityv_(n) can be related to the gradient of acoustic pressure as:

$\begin{matrix}{{v_{n}(x)} = {\frac{1}{i\;\rho\;{ck}}\frac{\partial p}{\partial n_{s}}(x)}} & (5)\end{matrix}$

where, ρ is the mass density of the medium. At this point, to use theintegral form of the Helmholtz equation in the analysis, an importantassumption is made that the field obeys Sommerfeld radiation condition,i.e.,

$r\left( {\frac{\partial p}{\partial n} - {i\;{kp}}} \right)$vanishes at infinity.

For the Fourier Transform Based NAH, selecting a Green's function (G)that satisfies the homogeneous boundary condition on the source surface,the integral form of Helmholtz equation can be written as:

$\begin{matrix}{{p(x)} = {{- \frac{1}{4\pi}}{\int{\int{{p\left( x_{s} \right)}\frac{\partial G}{\partial n}\left( {x,x_{s}} \right)d^{2}r_{s}}}}}} & (6)\end{matrix}$

This equation is known as the Rayleigh's first integral. If the sourcesurface is conformal to the hologram surface, then it is possible toconvolute the measured data using the known Green's function on thesource surface. The Green's function G(x, x_(S)) that satisfies thehomogeneous Dirichlet boundary condition on the plane z=z_(S) is givenin Maynard et al:

$\begin{matrix}{{G\left( {x,x_{s}} \right)} = {\frac{\exp\left\lbrack {i\; k\sqrt{\left( {x - x^{\prime}} \right)^{2} + \left( {y - y^{\prime}} \right)^{2} + \left( {z - z^{\prime}} \right)^{2}}} \right\rbrack}{\sqrt{\left( {x - x^{\prime}} \right)^{2} + \left( {y - y^{\prime}} \right)^{2} + \left( {z - z^{\prime}} \right)^{2}}} - \frac{\exp\left\lbrack {i\; k\sqrt{\left( {x - x^{\prime}} \right)^{2} + \left( {y - y^{\prime}} \right)^{2} + \left( {z + z^{\prime} - {2\; z_{S}^{\prime}}} \right)^{2}}} \right\rbrack}{\sqrt{\left( {x - x^{\prime}} \right)^{2} + \left( {y - y^{\prime}} \right)^{2} + \left( {z + z^{\prime} - {2\; z_{S}}} \right)^{2}}}}} & (7)\end{matrix}$

Therefore, the Rayleigh's first integral [equation (7)] can be writtenas,p(x,y,z)=∫∫_(−∞) ^(+∞) p(x′,y′,z _(S))G′(x−x′,y−y′,z−z _(S))dx′dy′  (8)

Here, x′, y′ are dummy variables for integration and G′ is the normalderivative of G. The measurement is available only on the hologramsurface H. This forms an inverse problem. Evaluating the integral at thehologram surface H:p(x,y,z _(H))=∫∫_(−∞) ^(+∞) p(x′,y′,z _(S))G′(x−x′,y−y′,z _(H) −z_(S))dx′dy′  (9)

This equation is in two dimensional convolution form, thus p(x, y,z_(S)) can be obtained using p(x, y, z_(H)). Two dimensional Fouriertransform can be used to convolute the equation. Symbol {circumflex over(p)} is used to denote two dimensional spatial Fourier transform.{circumflex over (p)}(k _(x) ,k _(y) ,z _(H))=∫∫_(−∞) ^(+∞) p(x,y,z_(H))e ^(i(k) ^(x) ^(k k′k) ^(y) ^(y))dxdy  (10)

Applying convolution on equations (9) and (8) and inverting:

$\begin{matrix}{{p\left( {x,y,z} \right)} = {\mathcal{F}^{- 1}\left\lbrack {{\hat{p}\left( {k_{x},k_{y},z_{H}} \right)}\left( \frac{{\hat{G}}^{\prime}\left( {k_{x},k_{y},{z - z_{S}}} \right)}{{\hat{G}}^{\prime}\left( {k_{x},k_{y},{z_{H} - z_{S}}} \right)} \right)} \right\rbrack}} & (11)\end{matrix}$

Note that the acoustic pressure at the source surface has beeneliminated in this equation. This equation (called reconstructionequation) gives the holographic reconstruction p(x, y, z) in threedimensions from the two dimensional measurements p(x, y, z_(H)). Thederivative of G can be calculated from equation (7). Therefore, the twodimensional Fourier transform of G′ is given by:Ĝ′(k _(x) ,k _(y) ,z)=e ^(iz)√{square root over (^(k) ² ^(. . . k) ^(x)² ^(. . . k) ^(y) ² )}  (12)

When k_(x) ²+k_(y) ²≤k², the kernel in the reconstruction formula, Ĝ′(k_(x),k_(y),z−z_(S)). Therefore, these waves are propagating waves andthey change only their phase (not the amplitude) while they propagate.When k_(x) ²+k_(y) ²>k², e^(−(z−zs))√{square root over (k_(x) ²+k_(y)²+k²)} is now a decaying exponential in z and the waves are referred toas evanescent. The boundary k_(x) ²+k_(y) ²>k² in k space is calledradiation circle. By inserting G′ in the holographic reconstructionformula,p(x,y,z)=y ⁻¹({circumflex over (p)}(k _(x) ,k _(y) ,z _(H))e ^(ik) ^(z)^((z . . . z) ^(dx) ⁾)  (13)

where, k_(z) is a function of k_(x) and k_(y): k_(z)=√{square root over(k²−k_(x) ²−k_(y) ²)}. The field gradient Δp is related to the particlevelocity field {tilde over (v)}(r) from equation (5). Taking gradient ofequation (13) the expression for the particle velocity components {tildeover (v)}_(x) in all the three directions, x=x,y,z can be obtained.

$\begin{matrix}{{{\overset{\sim}{v}}_{x}\left( {x,y,z} \right)} = {\frac{1}{\left( {2\pi} \right)^{2}\rho\; c}{\int{\int_{- \infty}^{+ \infty}{{{\hat{p}\left( {k_{x},k_{y},z_{H}} \right)}\left\lbrack {\left( {k_{x}/k} \right)e^{i\;{k_{z}{({z - z_{H}})}}}} \right\rbrack}e^{i\;{({{k_{x}x} + {k_{y}y}})}}\ d\; k_{s}d\; k_{y}}}}}} & (14)\end{matrix}$

The boundary element method (BEM) based NAH algorithm naturally suitsthe holography analysis (where only the boundary values are known)better than other numerical methods (such as finite element methods)because it reduces the problem dimension by one. Discretization isneeded only on the source surface. The solution to the Helmholtzequation (equation (4)) can be found using a surface integration as theGreen's function G(x, x_(S)) is known.

$\begin{matrix}{{\alpha\;{p(x)}} = {\int_{S_{k}}{\left( {{{p\left( x_{S} \right)}\frac{\partial}{\partial n_{S}}{G\left( {x,x_{S}} \right)}} - {{G\left( {x,x_{S}} \right)}\frac{\partial}{\partial n_{S}}{p\left( x_{S} \right)}}} \right)\ d\; S_{S}}}} & (15)\end{matrix}$

This equation is called Helmholtz integral equation. Here, x and x_(S)are the field and source points respectively. This equation is of theboundary integral form. The free-space Green's function G is given as:

$\begin{matrix}{{G\left( {x,x_{s}} \right)} = \frac{{\mathbb{e}}^{{\mathbb{i}}\;{kr}}}{4\pi\; r}} & (16)\end{matrix}$

The parameter α is used, among other reasons, to alleviate thesingularity at the corner points (or at the non-smooth nodes).

$a = \left\{ \begin{matrix}{1,} & {{x \in D},{{domain}\mspace{14mu}{not}\mspace{14mu}{including}\mspace{14mu}{source}\mspace{14mu}{surface}}} \\{{1/2},} & {{x \in S},{{smooth}\mspace{14mu}{source}\mspace{14mu}{surface}}} \\{{{\Omega/4}\pi},} & {{x \in S},{{non}\text{-}{smooth}\mspace{14mu}{source}\mspace{14mu}{surface}}}\end{matrix} \right.$

where, Ω is the solid angle. Changing the Helmholtz integral equation(15) to operator notation provides:

$\begin{matrix}{{\alpha\;{p(x)}} = {{({Dp})(x)} - {\left( {S\frac{\partial p}{\partial n_{s}}} \right)(x)}}} & (17)\end{matrix}$

The operators are given by the single layer potential,

${\left( {S\frac{\partial p}{\partial n_{s}}} \right)(x)} = {\int_{s_{s}}{\left( {{G\left( {x,x_{s}} \right)}\frac{\partial p}{\partial n_{s}}\left( x_{s} \right)} \right){\mathbb{d}S_{s}}}}$

and the double layer potential,

${({Dp})(x)} = {\int_{s_{s}}{\left( {{p\left( x_{s} \right)}\frac{\partial G}{\partial n_{s}}\left( {x,x_{s}} \right)} \right){\mathbb{d}S_{s}}}}$

Quadratic, quadrilateral elements are used in this study. The localcoordinates, ξ≡(ξ₁, ξ₂), are related to the global coordinates, x≡(x₁,x₂, x₃), by the transformation equation:

$\begin{matrix}{{{{x_{i}(\xi)} = {\sum\limits_{i = 1}^{L}{{N_{i}(\xi)}x_{i\; l}}}},{i = 1},2,{3;}}{L = 9}} & (18)\end{matrix}$

where, N_(i) are the shape functions; x_(il) is the i^(th) coordinate ofthe I^(th) node of the element. Isoparametric approximation is used inthis study. Therefore, any field can be approximated in the same way asthe coordinates. Therefore, the acoustic pressure at any point in them^(th) element p_(m) can be approximated as:

$\begin{matrix}{{{{p_{m}(\xi)} = {\sum\limits_{i = 1}^{L}{{N_{i}(\xi)}p_{ml}}}},{m = 1},2,\ldots\mspace{14mu},{M;}}{L = 9}} & (19)\end{matrix}$

where, p_(ml) is the acoustic pressure at the I^(th) node of the m^(th)element and M is the total number of elements. The normal derivative ofacoustic pressure (p_(n))_(m) at any point in the m^(th) elementproximated in a similar fashion:

$\begin{matrix}{{{{\left( p_{n} \right)_{m}(\xi)} = {\sum\limits_{i = 1}^{L}{{N_{i}(\xi)}\left( p_{n} \right)_{ml}}}},{m = 1},2,{{\ldots\mspace{14mu} M};}}{L = 9}} & (20)\end{matrix}$

The subscript n denotes the normal derivative. Now, the boundaryintegral equation (15) can be approximated as:

$\begin{matrix}\left. {\left. {\left. {\left. {{\alpha\;{p(x)}} = {\sum\limits_{m = 1}^{M}{\left\lbrack {\left( {\int_{\Delta\; S_{m}}^{\;}{\frac{\partial G}{\partial n_{S}}\left( {x,{x_{S}(\xi)}} \right){J(\xi)}}} \right){N(\xi)}} \right)\ {\mathbb{d}S_{S}}}}} \right)p_{m}} \right\rbrack - {\sum\limits_{m = 1}^{M}{\left\lbrack {\left( {\int_{\Delta\; S_{m}}^{\;}{{G\left( {x,{x_{S}(\xi)}} \right)}{J(\xi)}}} \right){N(\xi)}} \right){\mathbb{d}S_{S}}}}} \right)\left( p_{n} \right)} \right\rbrack & (21)\end{matrix}$

where, x_(s)(ξ) is calculated using equation (18), J(ξ) is the Jacobianof the transformation, ΔS_(m) is the element area, N(ξ) is a row vectorwith L=9 entries and p_(m) and (p_(n))_(m) are column vectors with L=9entries.

Two dimensional Gauss quadrature is used to approximate the elementalintegration. Legendre polynomial roots are used to define abscissas andweights of integration points as they are symmetric and appropriate forthe elements used here. If there are M elements and N nodes, thenequation (21) can be assembled in matrix form.

$\begin{matrix}{{\alpha\;{p(x)}} = {{\left( {\int_{\Delta\; S_{m}}^{\;}{\frac{\partial G}{\partial n_{S}}{JN}{\mathbb{d}S_{S}}}} \right)^{FS}p^{S}} - {\left( {\int_{\Delta\; S_{m}}^{\;}{{GJN}{\mathbb{d}S_{S}}}} \right)^{FS}p_{n}^{S}}}} & (22)\end{matrix}$

Or in operator notation:αp(x)=D ^(FS) p ^(S) −S ^(FS) p _(n) ^(S)  (23)

where, p^(S) and p_(n) ^(S) are column vectors of size N×1, D^(FS) andS^(FS) are row vectors of size 1×N, F is a field point, S is a sourcepoint, and FS is used here to denote the transformation.

Consider the field point is located at the hologram surface H. If thereare N_(H) measurement points in H, then each measurement point will havea row in the assembly matrix. Now, xϵD_(b), therefore α=1.p ^(H) =D ^(HS) p ^(S) −S ^(HS) p _(n) ^(S)  (24)

where, p^(S) and p_(n) ^(S) are column vectors of size N×1, p^(H) is acolumn vector of size N_(H)×1 and D^(FS) and S^(FS) are matrices of sizeN_(H)×N. Similarly, if the field points to be considered to be at thesource surface itself, then:αp ^(S) =D ^(SS) p ^(S) −s ^(SS) p _(n) ^(S)  (25)

If the surface is smooth, α=½,D ^(SS) p ^(S) =S ^(SS) p _(n) ^(S)  (26)

where {tilde over (D)}^(SS)=D^(SS)−α1. l is an identity matrix. Fromequations (23) and (26) p^(S) or p_(n) ^(S) can be eliminated. Eliminatep^(S) is done because the matrix D ^(SS) is usually better conditionedand sparse than S^(SS). After eliminating p^(S):[D ^(HS)(D ^(SS))⁻¹ S ^(SS) −S ^(HS) ]p _(n) ^(S) =p ^(H)  (27)Equation (27) is in the form ofAx=p.

If the matrix A is well-conditioned, then LU decomposition is used tosolve this system of equation to find out p_(n) ^(S). Then equation (26)can be used to find out p^(S). After this point, the problem is aforward problem to solve the acoustic pressure distribution at any othersurface.

Equivalent source model (ESM) is a rather easier method to implementwhen compared to the BEM algorithm. A term source strength q is definedas:p(x)=∫_(S) _(S) (G(x,x _(S))q/(x _(S)))dS _(S)  (28)

$\begin{matrix}{p_{i} = {\sum\limits_{j = 1}^{N}{G_{ij}q_{i}}}} & (29)\end{matrix}$

To avoid singularity in the reconstruction, the source strength isestimated on a virtual source surface instead of actual source surface.p _(H) =G ^(HV) q _(v)  (30)

Now, this source strength distribution on the virtual surface can beused to calculate the acoustic pressure distribution at the actualsource surface without the singularity problem.p ^(S) =G ^(VS) q ^(V)  (31)

The system Ax=p arrived using the BEM and ESM are usuallyill-conditioned especially when the matrix is too large. Therefore, themeasurement errors and the effects of evanescent waves will be amplifiedwhen the matrix A is inverted. The LU decomposition will result in verylarge coefficients in the inverted matrix. Therefore, the matrix needsto be regularized to reduce the singularities. In this kind ofsituations, the least square solutions by minimizing the system ofequations is the best approach. Consider singular value decomposition(SVD) of matrix A.A=UΣV ^(H)  (32)

The columns of U and V^(H) are left and right singular vectors. Σ is adiagonal matrix whose diagonal elements σ_(i) are called singularvalues. U and V are unitary matrices and the σ is a diagonal matrixwhich is very advantageous in computing their inverse individually.Therefore, the pseudo-inverse, A^(†), can be written as:A ^(†) =VΣ ^(†) U ^(H)  (33)

There are systematic methods to alleviate the singularity in largeill-conditioned systems (details can be seen in the book by Hansen[17]). The regularized solution x_(y) is given by:

$\begin{matrix}{x_{\gamma} = {{{VF}{\sum^{\dagger}{U^{H}p}}} = {\sum\limits_{i = 1}^{R}{{f_{i}\left( \frac{u_{i}^{H}p}{\sigma_{i}} \right)}v_{i}}}}} & (34)\end{matrix}$

where, the diagonal matrix F with diagonal entries f_(i) acts as afilter on the singular matrix. The filter can be chosen in many ways assuggested in prior research. Tikhonov regularization is awell-established direct method and was used in the examples providedbelow. This method defines the filter factor in terms of a singleparameter, γ, as:

$\begin{matrix}{f_{i} = \frac{\sigma_{i}^{2}}{\sigma_{i}^{2} + \gamma}} & (35)\end{matrix}$

By choosing a right value for the parameter γ, the errors in thesolution can be minimized. However, this is done at the cost of losingpart of the physical meaning of the data. There are methods to choosethe parameter systematically. The three popular methods for theparameter choice are Morozov discrepancy principle, Generalized crossvalidation (GCV), and L-curve.

GCV is the suitable method when the error in the measurement p is notknown exactly.

$\begin{matrix}{{= \frac{{{{Ax}_{\gamma} - p}}_{2}^{2}}{N_{H} - \mu}}{{where},{\mu = {\sum\limits_{i = 1}^{R}{f_{i}.}}}}} & (36)\end{matrix}$γ is optimum when G is minimum.

In an implementation of the present invention, NAH is used to measureboth the acoustic pressure and the acoustic velocity on the outside ofthe enclosure. This is a 2-D measurement that estimates the pressure andvelocity at various spots along the enclosure using a microphone array.Thus spatial variation is measured without having to physically move thesensor over the entire enclosure. The pressure is measured inside theenclosure and thus a spatial varying pressure difference can beestimated along with a spatially varying sound velocity. The ratio ofsurface pressure to velocity is used to separate sound being transmittedthrough the structure from sound travelling through cracks. Soundtraveling through cracks has a much lower ratio of pressure to velocityand so our method can be used to only analyze the sound travelingthrough cracks as well as identify the location of the cracks. Thismeasurement is done at several frequencies.

The ratio of the pressure difference between interior and exterior tothe acoustic velocity in the crack regions is the acoustic transferimpedance of the crack. From the frequency dependent acoustic transferimpedance, the leakage properties of the cracks are estimated. Thisconversion is described below.

In one implementation, the leakage of the individual cracks is addedtogether to get the total leakage for the section under test.

In one implementation, a key difference in the methodology is theestimation of the acoustic velocity in addition to the acoustic pressureon the outside of the enclosure. It is the combination of having theacoustic velocity and the interior-to-exterior pressure difference thatallows one to isolate leaks with a single measurement at a distance andto get the transfer impedance that allows one to estimate the leakageinformation without resorting to correlation analysis. The microphonearray should generally be located as close as possible to the buildingenclosure and must be located within half a wavelength of the surfacewhere the wavelength of the sound at the frequency under test is theratio of the speed of sound to frequency.

The acoustic velocity and pressure can be measured in a number ofdifferent manners. One embodiment of measurement at a distance is theNearfield Acoustic Holography, but there are other methods that could beused including the use of a sound intensity probe that is used tomeasure sound intensity across the outside of the enclosure. One couldalso use acoustic beamforming to measure the surface pressure and usesome other technology to measure the acoustic velocity

ABMIS can be used to measure small sections of eventual sealedstructures (buildings, aircraft fuselages, hazardous containmentenclosures, etc.) during construction to resolve infiltration issuesearlier and subsequently at much lower cost/impact to the build effort.This previously unavailable infiltration-detection (and subsequentavoidance) technique could be very valuable in the construction ofrelatively small sealed structures where infiltration has high costnegative impact (failure, exposure, leakage, etc.). It is clearly farsuperior to the two techniques currently employed in industry(pressurization testing and tracer gas testing) that are inefficient,inaccurate, and can only be used in finished/sealed structures.

Acquiring Acoustic Properties

Once the acoustic velocity and pressure are measured, they can be usedto acquire other acoustic properties necessary to determining theinfiltration airflow Q per equation (1). Therefore, infiltration flow itis necessary to calculated the infiltration flow coefficient CL as n theinfiltration exponent. A general background is necessary with regard tohow cracks are modeled or considered in regard to building infiltrationstudy. In the present invention, velocity and pressure are the knownquantities.

Airflow through a crack is governed by the Navier-Stokes equations forNewtonian fluids (Fox, McDonald, & Pritchard, 2004):

$\begin{matrix}{{\frac{\partial\overset{\rightarrow}{v}}{\partial t} + {\left( {\overset{\rightarrow}{v} \cdot \overset{\rightarrow}{\nabla}} \right)\overset{\rightarrow}{v}}} = {{{- \frac{1}{\rho}}{\overset{\rightarrow}{\nabla}p}} + {\mu{\nabla^{2}\overset{\rightarrow}{v}}}}} & (37)\end{matrix}$where the fluid density is given by ρ, the fluid velocity is given by{right arrow over (v)}, the fluid pressure is given by p, and theviscosity is given by μ. If steady flow is assume, velocity will notvary in time and Eqn. (37) reduces to

$\begin{matrix}{{\left( {\overset{\rightarrow}{v} \cdot \overset{\rightarrow}{\nabla}} \right)\overset{\rightarrow}{v}} = {{{- \frac{1}{\rho}}{\overset{\rightarrow}{\nabla}p}} + {\mu{{\nabla^{2}\overset{\rightarrow}{v}}.}}}} & (38)\end{matrix}$Conservation of mass also requires that

$\begin{matrix}{{\frac{\partial\rho}{\partial t} + {\overset{\rightarrow}{\nabla}{\cdot \left( {\rho\overset{\rightarrow}{v}} \right)}}} = 0.} & (39)\end{matrix}$

If the flow is assumed to be incompressible, which is appropriate forinfiltration, then density will not change in time, and so Eqn. (39)reduces to{right arrow over (∇)}·{right arrow over (v)}=0.  (40)Eqn (38) and (40) are the starting point for our analysis. The velocityfield can be broken into components and written as{right arrow over (v)}=v _(x) {circumflex over (x)}+v _(y) ŷ+v _(z){circumflex over (z)}.  (41)To understand the relationship between pressure drop across a crack andthe flow through a crack, consider the simplified geometry shown in FIG.1A a crack has a length L, a height H, and a depth D. For all analysisthat follows, D>>H.

In breaking down a crack into simpler geometries, it can be considered,where height is significantly less than the length and depth, to be acombination of an inlet, a section of Poiseuille flow and an outlet.Examination of two limiting cases is useful in understanding the typicalrange of values of the exponent n in Eqn. (1). The first case is fullydeveloped flow where the length and depth of the crack are extremelylarge compared to the crack height H, commonly called Poiseulle flowwhich yields n=1.0. The second case is flow through a thin orifice, theinlet and the outlet, which yields n=0.5. One can assume that thepressure gradient driving the flow is a gradient only in the x directionwhich means that

$\begin{matrix}{{\nabla p} = {\frac{\partial p}{\partial x}.}} & (42)\end{matrix}$

A first step in determining the infiltration exponent is to determinethe portion of the crack assumed to be Poisuell flow. Thus, when thedepth of the crack D and length of the crack L are much larger than theheight H and the flow is primarily in the x direction. Assuming thelength of the crack L and depth D are is infinite, flow will be fullydeveloped in the x direction and there will be no variation in the ydirection. By symmetry arguments, the velocity will only vary as afunction of z direction and thus the velocity field reduces to{right arrow over (v)}=v _(x)(z){circumflex over (x)}.  (43)With the velocity in this form,

$\begin{matrix}{{\left( {\overset{\rightarrow}{v} \cdot \overset{\rightarrow}{\nabla}} \right)\overset{\rightarrow}{v}} = {{{v_{x}(z)}\frac{\partial}{\partial x}{v_{x}(z)}} = 0}} & (44)\end{matrix}$and thus Eqn (38) reduces to

$\begin{matrix}{{\frac{1}{\rho}\frac{\partial p}{\partial x}} = {\mu{\frac{\partial^{2}{v_{x}(z)}}{\partial z^{2}}.}}} & (45)\end{matrix}$Eqn. (45) can then be integrated twice in z to yield

$\begin{matrix}{{v_{x}(z)} = {{\frac{1}{2{\rho\mu}}\frac{d\; p}{d\; x}z^{2}} + {C_{1}z} + {C_{2}.}}} & (46)\end{matrix}$Using the no-slip boundary conditions (v_(x)=0) on the walls of thecrack (z=0 and z=H), the constants can be evaluated to yield

$\begin{matrix}{{v_{x}(z)} = {\frac{1}{2{\rho\mu}}\frac{d\; p}{d\; x}{\left( {z^{2} - {Hz}} \right).}}} & (47)\end{matrix}$The total volume flow rate of fluid through the crack, Q, is given byintegrating the velocity across the crack face

$\begin{matrix}{{Q = {{\underset{y,z}{\int\int}v_{x}d\;{yd}\; z} = {{\int_{0}^{D}{\int_{0}^{H}{\frac{1}{2{\rho\mu}}\ \frac{d\; p}{d\; x}\left( {z^{2} - {Hz}} \right)d\;{yd}\; z}}} = {{- \frac{{DH}^{3}}{12{\rho\mu}}}{\frac{d\; p}{d\; x}.}}}}}\ } & (48)\end{matrix}$

For a given pressure difference across the crack, Δp, the pressuregradient is given by

$\begin{matrix}{{\frac{\partial p}{\partial x} = {- \frac{\Delta\; p}{L}}},} & (49)\end{matrix}$with the negative sign appearing because the pressure gradient isopposite the actual direction of flow. Combining Eqn. (48) and Eqn. (49)provides that the pressure drop created by viscous flow Q through athrough a long crack, ignoring end effects, will be

$\begin{matrix}{{\Delta\; p} = {\frac{12{\rho\mu}\; L}{{DH}^{3}}{Q.}}} & (50)\end{matrix}$Eqn. (50) shows the flow exponent, n, and flow coefficient, C_(L), toput this into the power law form of Eqn. (1) are given by

$\begin{matrix}{{n = 1}{C_{L} = {\frac{{DH}^{3}}{12{\rho\mu}\; L}.}}} & (51)\end{matrix}$

As a second step, to understand the pressure drop flow relation at theinlet (and by correlation the outlet), the same assumption is made thatthe inlet (and outlet) is of a crack that is very long and very deep. Across sectional diagram is shown in FIG. 1B. Well away from the inlet,the pressure is p₀, and the flow velocity is essentially zero. Justwithin the crack, the flow velocity has increased to v₁ and the pressurehas dropped to p₁. Over a short distance, viscous losses are negligibleand the pressure drop will be completely from the force required toaccelerate the fluid from a velocity of 0 to v₁.

While the flow into the inlet is quite complex, it still followsBernoulli's theorem, a form of conservation of energy, which states thatalong any streamline the quantity

$\begin{matrix}{{\frac{1}{2}{\overset{\rightarrow}{v}}^{2}} + \frac{p}{\rho}} & (52)\end{matrix}$is constant. Applying Eqn. (52) to a streamline that starts far from thecrack where the flow velocity is zero and the pressure is p=p₀ and theflow accelerates to a velocity v₁ and a pressure p₁ inside the inletyields the equation

$\begin{matrix}{\frac{p_{0}}{\rho} = {{\frac{1}{2}v_{1}^{2}} + {\frac{p_{1}}{\rho}.}}} & (53)\end{matrix}$Identifying Δp=p₀−p₁ and noting that the volume rate of flow will beQ=HDv₁ one can write

$\begin{matrix}{{\Delta\; p} = {{\frac{1}{2}\rho\; v_{1}^{2}} = {\frac{1}{2}{\frac{\rho\; Q^{2}}{HD}.}}}} & (54)\end{matrix}$This equation can be inverted to yield

$\begin{matrix}{{Q = {\sqrt{\frac{2}{\rho}}{HD}\;\Delta\; p^{0.5}}},} & (55)\end{matrix}$from which the flow exponent, n, and flow coefficient, C_(L), are givenby

$\begin{matrix}{{n = 0.5}{C_{L} = {{DH}{\sqrt{\frac{\Delta\; p}{\rho}}.}}}} & (56)\end{matrix}$An outlet has essentially the same physics as an inlet and byBernoulli's theorem will end up having the same relationshippressure-flow as Eqn. (54).

Having determined the flow exponent for the components of the crack aswell as the flow coefficient (as a function of other parameters of thesystem), a simple model for a complete crack can be considered bycombination of an inlet, outlet, and a section of Poiseuille flow aseach was described above. Thus using Eqn. (54) and Eqn. (50) a firstapproximation to the relation between pressure drop and volume of flowthrough a crack will be

$\begin{matrix}{{\Delta\; p} = {{{\frac{1}{2}\frac{\rho\; Q^{2}}{D^{2}H^{2}}} + {\frac{12{\rho\mu}\; L}{{DH}^{3}}Q} + {\frac{1}{2}\frac{\rho\; Q^{2}}{D^{2}H^{2}}}} = {{\frac{12{\rho\mu}\; L}{{DH}^{3}}Q} + {\frac{\rho\; Q^{2}}{D^{2}H^{2}}.}}}} & (57)\end{matrix}$

Because of the approximations that were made to the inlet and outletflow and the assumption of fully developed flow within the crack,modifications to Eqn. (57) are necessary. A more general form of Eqn.(57) introduced by Walker (Walker, Wilson, & Sherman, 1997) isΔp=AQ+BQ ²  (58)where the coefficients A and B are typically developed through analyticanalysis (as above), computational analysis (often using computationalfluid dynamics), or measurements. Comparison of Eqns 57 and 58 showsthat for our simple slit, the coefficients and A and B are

$\begin{matrix}{{A = \frac{12{\rho\mu}\; L}{{SH}^{2}}}{and}{B = \frac{\rho}{S^{2}}}} & (59)\end{matrix}$where S=DH is the cross sectional area of the crack.

Eqn. 58 is not in the power law form of Eqn. 1. It has been posited byprior researchers that the power law form is more useful, especiallywhen ambient temperature and pressure corrections need to be made. Whenmeasured data at several different pressures are available, Eqn. 58 canbe cast into a power law form of Eqn. 1 through a least squares fit ofEqn. 1 to Eqn. 58 for C_(L) and n.

In addition to the above method using C_(L) and n can be estimatedthrough other means, in particular, if the alternative form of Q(Δp) isknown, one can estimate n in the following manner:

$\begin{matrix}{Q = {C_{L}\left( {\Delta\; P} \right)}^{n}} & (60) \\{\frac{dQ}{d\;\Delta\; p} = {{{nC}\left( {\Delta\; p} \right)}^{n - 1} = {n\frac{Q}{\Delta\; p}}}} & (61)\end{matrix}$and therefore

$\begin{matrix}{n = {\frac{dQ}{d\;\Delta\; p}{\frac{\Delta\; p}{Q}.}}} & (62)\end{matrix}$Inverting Eqn. (58) provides an alternative equation for Q and terms ofΔp

$\begin{matrix}{Q = {\frac{\sqrt{A^{2} + {2B\;\Delta\; p}} - A}{2B}.}} & (63)\end{matrix}$Using Eqn. 62 in Eqn. 61 and simplifying provides:

$\begin{matrix}{n = {\frac{1}{2}{\left( {\frac{A}{\sqrt{A^{2} + {4B\;\Delta\; p}}} + 1} \right).}}} & (64)\end{matrix}$Eqn. 63 is not independent of Δp, but such is not seen in the limits ofPoiseuille flow (B→0), n→1 and in the limit of inlet dominated flow(A→0), n→0.5 as expected. Using Eqns 28 and 24:

$\begin{matrix}{C_{L} = {\Delta\; p^{- n}{\frac{\sqrt{A^{2} + {4B\;\Delta\; p}} - A}{2B}.}}} & (65)\end{matrix}$

If one can find the coefficients A and B, the flow coefficient C_(L) andflow exponent n can be estimated by use of a standardized referencepressure difference.

The coefficients A and B can be estimated from acoustic impedance Theacoustic impedance of slits and cracks has been studied by many authors(Sivian, 1935; Wood & Thurston, 1953) and can be calculated. Theacoustic transfer impedance of a rectangular slit of length L, height H,depth D, and cross sectional area S=DH at a frequency f is given by

$\begin{matrix}{Z = {\frac{j\;{\omega\rho}\; L}{S}\frac{\left( {1 + j} \right)Y}{{\tan\left( {\left( {1 + j} \right)Y} \right)} - {\left( {1 + j} \right)Y}}}} & (66)\end{matrix}$where =√{square root over (−1)}, ω=2πf, and the ratio of slit thicknessto viscous penetration depth, Y, is given by

$\begin{matrix}{Y = {H{\sqrt{\frac{\omega\rho}{8\mu}}.}}} & (67)\end{matrix}$The low frequency (ω→0) limit of Eqn. 68 is

$\begin{matrix}{Z_{0} = {{Z❘_{\omega\rightarrow 0}} = {{12\frac{\mu\; L}{{SH}^{2}}} + {j\;\omega\frac{6\rho\; L}{5S}}}}} & (68)\end{matrix}$The high frequency (ω→∞) limit of Eqn. 68 is

$\begin{matrix}{Z_{\infty} = {Z❘_{\omega\rightarrow\infty}{{\sqrt{2{\mu\rho\omega}}\frac{L}{SH}} + {j\;\omega\frac{\rho\; L}{S}}}}} & (69)\end{matrix}$The real part of Eqn. 70 can be identified directly as the infiltrationcoefficient A and can thus be measured directly from a low frequencyacoustic impedance measurementA=Re{Z| _(ω→0)}.  (70)

The coefficient B=μ/S² depends only on ρ and crack cross-sectional areaS and so if an estimate of S is available, the coefficient B can bedirectly estimated. The area S could be estimated from analysis of thesurface acoustic measurements made using acoustic holography above.

The coefficient B=μ/S², cannot be directly identified from impedancemeasurements; however, if one is able to estimate the crack length L bysome other means, such as time delay analysis of reflections in thecrack or crack cavity resonance analysis (Munjal, 1987), then

$\begin{matrix}{B = {\frac{25}{36\rho\; L^{2}}{Im}{\left\{ {Z❘_{\omega\rightarrow\infty}} \right\}^{2}.}}} & (71)\end{matrix}$

The coefficients A and B were estimated directly from impedancemeasurements through comparison of analytics expressions developed fromidealizations. Real world cracks will tend to be more complicated andthe analytic expressions may not be very accurate. Alternatively, onecan develop better relationships between the static flow coefficientsand acoustic impedance through correlations of static flow relationshipsand acoustic impedance as calculated through computational fluiddynamics (CFD).

The analysis above does show that A is closely related to the lowfrequency limit of the real part of the transfer impedance and B can berelated to the imaginary part of the transfer impedance. Correlationcoefficients relating the transfer impedance computed through CFD,Z_(CFD)(ω), and the coefficients A_(CFD) and B_(CFD) computed by CFD canbe defined as

$\begin{matrix}{{{CC}_{A} = \frac{A_{CFD}}{\int\limits_{\omega_{\min}}^{\omega_{\max}}{{Re}\left\{ {Z_{CFD}(\omega)} \right\} d\;\omega}}}{and}} & (72) \\{{CC}_{B} = \frac{B_{CFD}}{\int\limits_{\omega_{\min}}^{\omega_{\max}}{{Im}\left\{ {Z_{CFD}(\omega)} \right\} d\;\omega}}} & (73)\end{matrix}$where ω_(min) and ω_(max) are selected by the user to match the actualrange of frequencies used to measure the transfer impedance, Z_(m).Then, the coefficients of A and B of the actual cracks are found as

$\begin{matrix}{{A = {{CC}_{A}{\int\limits_{\omega_{\min}}^{\omega_{\max}}{{Re}\left\{ {Z_{m}(\omega)} \right\} d\;\omega}}}}{and}} & (74) \\{B = {{CC}_{B}{\int\limits_{\omega_{\min}}^{\omega_{\max}}{{Im}\left\{ {Z_{m}(\omega)} \right\} d\;{\omega.}}}}} & (75)\end{matrix}$

EXAMPLES

This section describes experiments regarding certain implementations ofthe present invention. To qualify various algorithms of NAH aloudspeaker experiment was designed. A sub-woofer of diameter 10 incheswas placed in an anechoic chamber located at IIT Fluid Dynamics ResearchCenter. Simultaneous measurement at all the points in the hologram isnot necessary for the frequency domain NAH analysis that are used inthis study. Therefore, a single microphone fixed on a motorized traverse(shown in FIG. 2(a)) is used to record the sound signal on a twodimensional surface (which is called hologram surface). The hologramsurface is located at distance 0.5 inches from the source. A quarterinch, B & K (model 4338) microphone is used here with appropriate powersupply and filter. 204800 samples are acquired using an NI DAQ module ata rate of 100 kHz. The measurement aperture is 28 in×28 in rectanglewith 113×113 measurement points (0.25 in between adjacent measurementpoints). A sine signal of single frequency is generated in Labview andsupplied through an NI DAC to a power amplifier. The amplified signalruns the loudspeaker. Frequencies considered for this test are 317 Hz,1098 Hz, and 5004 Hz.

For the leakage detection experiment, a wooden model of a building isbuilt with the dimensions 68″×50″×32″. This building model has a 20″×6″opening which can hold an insert plate with specified slots on it. Theseslots can be considered as known leakage areas. The loudspeaker ismounted inside the model. The measurement is performed in the same wayas the loudspeaker experiment. The hologram surface is 0.5 in away fromthe building surface. This setup is shown in FIG. 2b ). FIG. 4 shows thetwo different insert plates used in these examples. The insert platesare of dimensions 20″×6″ and made up of acrylic glass material. Theinsert I has a rectangular slot and an elliptical slot each of width 6in and maximum height of 0.25 in. Insert II has circular holes ofvarious diameters ranging from 2 in to 1/16 in.

The computer programs for FFT, BEM and ESM based methods were developedin FORTRAN and compiled using GNUFORTRAN compiler. The results wereplotted using MATLAB.

As mentioned in the previous section, to check the algorithms aloudspeaker experiment was used. The acoustic pressure measured on thehologram is used to reconstruct the acoustic pressure and particlevelocity field on the source surface. FIG. 4 shows the contours ofacoustic pressure on the source surface (which is a planar surface veryclose the speaker surface) for the input frequency of 5004 Hz. Theresults show that all the three algorithms are able to detect thesources reasonably well. The predicted pressure levels are a littledifferent for each algorithm due to the different numericalapproximations used in them. However, when converted to log (dB) scalethese differences would not make so much sense. For the other twofrequencies, the reconstruction results were similar (not shown here).

The dynamic range for all the plots shown in this section are chosen as65 percent of the maximum value. The calculated acoustic pressure andparticle velocity are complex quantities. Therefore, the absolute valueis used here to plot.

The sound leaks through the same locations as the air. Therefore, soundis generated inside the building model and the acoustic pressure ismeasured on hologram surface and it is inverse reconstructed to thebuilding model surface to detect the known leakage locations.Diffraction characteristics depends on the frequency/wavelength of thepropagating sound wave. Also, the NAH works very well when the areasdominated by evanescent waves are minimum (the evanescent cut-off isalso based on frequency/wave number). Therefore, three different inputfrequencies are chosen for this study (5004, 1098 and 317 Hz) and theresults are presented below.

Higher Input Frequency

FIG. 5 shows the acoustic pressure fields on the surface of insert Icalculated using various algorithms. The FFT algorithm is based on theRayleigh's integral and the ESM theory neglects the dipole termcompletely. Therefore, these two algorithms are supposed to be inferiorto the BEM algorithm in terms of accuracy. It is clear from FIG. 5 thatBEM gives the closer idea of the leakage location. However, the solutionlacks smoothness compared to the ESM and FFT solutions. Thisnon-smoothness in the solution is expected when using the quadraticapproximation in the BEM algorithm as explained in the prior art.However, when using linear approximation the solution could loseaccuracy. FIG. 6 shows the acoustic pressure field reconstructed on thesurface of insert II. All the algorithms are able to detect the largersource. However, none of the algorithms are able to resolve the smallersources on the insert. This could be due to the domination of the largersource in the field. Another set of experiments must be conducted withsmaller sources exclusively in the absence of larger sources to find outif at all a given algorithm is able to detect a source of particularsize.

FIGS. 7 and 8 show the calculated particle velocity field on insert Iand II respectively. The contours looks similar to the acoustic pressurecontours shown in FIGS. 5 and 6 because the particle velocities arebasically calculated using the acoustic pressure field.

Lower Input Frequency

The lower frequency chosen in this study is 317 Hz. FIGS. 9 and 10 showthe acoustic pressure distribution calculated using various algorithmson the surfaces of insert I and insert II respectively. The localizingeffect is much worse for this frequency when compared with the higherfrequency. NAH methods have inherent problem with the evanescent waves.For the lower frequency analysis, most of the domain contains evanescentcomponents when compared to the higher frequency analysis. Therefore, itis confirmed that the higher the wave number, better the reconstructedresult. FIGS. 11 and 12 shows the particle velocity distribution for thesame frequency. One can easily notice that the particle velocity valuesare not realistic. These higher values of pressure and particlevelocities appear in the NAH calculation because the errors in themeasurement grow exponentially during inverse calculation when thedomain is dominated by evanescent waves. There are ways to reduce thiseffect which will be discussed below.

Intermediate Input Frequency

Intermediate frequency case (1098 Hz) response is in between higher andlower frequency cases. For this case, not all the results are shownhere. A sample can be seen in FIG. 13. FIG. 13 shows the acousticpressure and particle velocity distribution on the surface of insert II.The shown result is obtained using ESM algorithm.

Effect of Regularization and Resolution

As mentioned earlier, the evanescent components can be minimized in thecalculation using various techniques. The regularization technique iswell known in some cases to alleviate this error. FIG. 14 shows theregularized ESM result on the surface of insert I for the lowerfrequency case. GCV method is used here to select the Tikhonovparameter. It is well known that sometimes GCV over-predicts theregularization parameter. In the present case also it is clear that theGCV applies over-regularization. Therefore, the pressure and particlevelocity values are predicted very low. Also, the localization effect islost in the contour.

Bai suggests reducing the resolution increases the accuracy ofreconstructed values. The resolution is suggested to be double thedistance between hologram and source. However, due to the scale of thepresent model, this suggested resolution is not desirable. Therefore,the resolution is reduced from Δx=0.25 in to Δx=0.5 in and the sampleresults can be found in FIG. 15. FIG. 15 shows the particle velocitydistribution on the surface of insert I. The values are more realisticthan the higher resolution results.

From the results presented above, it is evident that the NAH algorithmshave the potential to be used in the future to detect leakages. Higherinput frequencies give better results because of their higher wavenumbers as evanescent waves appear only in the regions where k²<k_(x)²+k_(y) ². FFT and ESM algorithms arrive at similar results in most ofthe cases. The BEM results are nonsmooth because of the quadraticapproximation used. Tikhonov regularization could be an effective methodto reduce the error by avoiding some of the singular values. However,choosing the proper Tikhonov parameter is a difficult task andgeneralized cross validation is not performing very well for the presentgeometry. Therefore, modern methods like L-curve criterion must be triedfor this geometry in the future. Reducing the resolution usually helpsto increase the accuracy and the present results confirm that.

The foregoing description of illustrative embodiments has been presentedfor purposes of illustration and of description. It is not intended tobe exhaustive or limiting with respect to the precise form disclosed,and modifications and variations are possible in light of the aboveteachings or may be acquired from practice of the disclosed embodiments.It is intended that the scope of the invention be defined by the claimsappended hereto and their equivalents.

What is claimed is:
 1. A method of detecting leakage comprising:emitting sound at a frequency and having a wavelength from anelectroacoustic transducer positioned within a structure; measuring,with a microphone positioned internal to the structure, an internalpressure of the emitted sound within the structure; positioning aholographic array of microphones external to the structure, the array ofmicrophones positioned, relative to the structure, within half of thewavelength of the emitted sound; measuring, by nearfield acousticholography with the array of microphones, external pressure and externalvelocity of the sound at discrete points on the exterior of thestructure; determining at each of the points an interior-to-exterioracoustic transfer impedance; and identifying, based upon the determinedacoustic transfer impedance, the size and location of a leak in thestructure.
 2. The method of claim 1, wherein detecting sound comprisesdetecting sound by nearfield acoustic holography.
 3. The method of claim2, wherein detecting sound by nearfield acoustic holography comprisespositioning a holographic array comprising a plurality of microphonespositioned external to the structure.
 4. The method of claim 3 whereindetermining the leak comprises applying a nearfield acoustic holographyalgorithm selected from the group consisting of Fourier transform (FFT)based NAH, Boundary element method (BEM) based NAH, and Equivalentsource model (ESM) based NAH.
 5. The method of claim 1, whereindetecting sound comprises detecting sound by a sound intensity probepositioned external to the structure.
 6. The method of claim 1, whereinexternal pressure is measured by acoustic beamforming.
 7. The method ofclaim 1, further comprising determining an acoustic transfer impedanceof the leak.
 8. The method of claim 1, wherein determining the leak inthe structure comprises determining at least one acoustic property basedupon the measured internal pressure, external pressure, and velocity. 9.The method of claim 1, comprising determining a plurality of leaks. 10.The method of claim 9, further comprising determining the leakage for asection of the structure by adding the determined plurality of leakscorresponding to the section.
 11. The method of claim 1, whereindetermining a leak comprises determining the ratio of pressure tovelocity for a plurality of sections.
 12. The method of claim 1, whereinthe emitted sound is within the range of 100 Hz to 5 kHz and at least 60dB.
 13. A system for detecting leakage of a structure, comprising: anelectroacoustic transducer positioned inside the structure andconfigured to emit an acoustic wave having a phase, frequency andamplitude; an interior transducer located inside the structure and incommunication with a computer to provide the phase, frequency, andamplitude of the emitted acoustic wave; and an external acoustictransducer system configured to detect acoustic pressure and acousticvelocity on the outside of the structure within a distance of half of awavelength of the emitted acoustic wave.
 14. The system of claim 13,wherein the acoustic transducer system comprises an acoustic holographymicrophone array configured to receive the acoustic waves on the outsideof the structure and in communication with the computer.
 15. The systemof claim 14, wherein the acoustic holography array comprises an array ofpressure transducers.
 16. The system of claim 14, wherein the acousticholography array is a nearfield acoustic holography array.
 17. Acomputer system for leakage detection comprising: a loudspeakerconfigured to emit an acoustic wave having a phase, frequency andamplitude within a structure, an internal microphone positioned withinthe structure and in communication with a computer to provide the phase,frequency, and amplitude of the emitted acoustic wave; a microphonearray positioned outside of the structure within half of a wavelength ofthe emitted acoustic wave and configured to receive the acoustic wave,the microphone array in communication with the computer providingacoustic pressure data to the computer; the computer including aprocessor and a tangible computer-readable medium operatively connectedto the processor and including computer code configured to: determineleakage from the array acoustic velocity and pressure data associatedwith each of a plurality of points.
 18. The system of claim 17, whereinthe microphone array comprises an array of pressure transducers.
 19. Thesystem of claim 17, wherein the microphone array is a nearfield acousticholography array.